{
  "id": "2102.03144",
  "version": "v1",
  "published": "2021-02-05T12:49:25.000Z",
  "updated": "2021-02-05T12:49:25.000Z",
  "title": "Spanning trees in dense directed graphs",
  "authors": [
    "Amarja Kathapurkar",
    "Richard Montgomery"
  ],
  "comment": "21 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In 2001, Koml\\'os, S\\'ark\\\"ozy and Szemer\\'edi proved that, for each $\\alpha>0$, there is some $c>0$ and $n_0$ such that, if $n\\geq n_0$, then every $n$-vertex graph with minimum degree at least $(1/2+\\alpha)n$ contains a copy of every $n$-vertex tree with maximum degree at most $cn/\\log n$. We prove the corresponding result for directed graphs. That is, for each $\\alpha>0$, there is some $c>0$ and $n_0$ such that, if $n\\geq n_0$, then every $n$-vertex directed graph with minimum semi-degree at least $(1/2+\\alpha)n$ contains a copy of every $n$-vertex oriented tree whose underlying maximum degree is at most $cn/\\log n$. As with Koml\\'os, S\\'ark\\\"ozy and Szemer\\'edi's theorem, this is tight up to the value of $c$. Our result improves a recent result of Mycroft and Naia, which requires the oriented trees to have underlying maximum degree at most $\\Delta$, for any constant $\\Delta$ and sufficiently large $n$. In contrast to the previous work on spanning trees in dense directed or undirected graphs, our methods do not use Szemer\\'edi's regularity lemma.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-05T12:49:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C20",
      "05C05",
      "05C35"
    ],
    "keywords": [
      "dense directed graphs",
      "spanning trees",
      "maximum degree",
      "szemeredis regularity lemma",
      "vertex graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}